A ug 2 00 5 Las Vergnas cube conjecture and reconstruction properties of the cube matroid February 2 , 2008 Ilda
نویسنده
چکیده
Las Vergnas Cube Conjecture states that the cube matroid has exactly one class of orientations. We prove that this conjecture is equivalent to saying that the oriented matroid Aff(C), of the affine dependencies of the n-cube C := {−1, 1} over IR, can be reconstructed from the underlying matroid and one of the following partial lists of signed circuits or cocircuits: 1) the signed circuits of rank 3 or 2) the positive signed cocircuits.
منابع مشابه
Orientability of Cubes and Las Vergnas Cube Conjecture
We introduce a class of combinatorial cubes, cubic matroids, and study orientability of the matroids in this class. The results obtained give, in particular, a new insight on Las Vergnas Cube Conjecture.
متن کاملSymmetry, oriented matroids and two conjectures of Michel Las Vergnas
The paper has two parts. In the first part we survey the existing results on the cube conjecture of Las Vergnas. This conjecture claims that the orientation of the matroid of the cube is determined by the symmetries of the underlying matroid. The second part deals with euclidean representations of matroids as geometric simplicial complexes defined by symmetry properties abstracting those of zon...
متن کاملOn a Mutation Problem for Oriented Matroids
For uniform oriented matroids M with n elements, there is in the realizable case a sharp lower bound Lr (n) for the number mut(M) of mutations ofM : Lr (n) = n ≤ mut(M), see Shannon [17]. Finding a sharp lower bound L(n) ≤ mut(M) in the non-realizable case is an open problem for rank d ≥ 4. Las Vergnas [9] conjectured that 1 ≤ L(n). We study in this article the rank 4 case. RichterGebert [11] s...
متن کاملThe switching game on unions of oriented matroids
In 1986, Hamidoune and Las Vergnas [3] introduced an oriented matroid version of the so-called Shannon’s switching game. They conjectured that the classification of the directed switching game on oriented matroids is identical to the classification of the non-oriented version. In this note, we support this conjecture by showing its validity for an infinity class of oriented matroids obtained as...
متن کاملThe Las Vergnas polynomial for embedded graphs
The Las Vergnas polynomial is an extension of the Tutte polynomial to cellularly embedded graphs. It was introduced by Michel Las Vergnas in 1978 as special case of his Tutte polynomial of a morphism of matroids. While the general Tutte polynomial of a morphism of matroids has a complete set of deletion-contraction relations, its specialisation to cellularly embedded graphs does not. Here we ex...
متن کامل